Asymptotic-diffusion analysis of retrial queue with two-way communication and unreliable server

In this paper Markovian retrial queue with multiple types of outgoing calls and unreliable server, which could be used as a mathematical model of a call center is considered. Incoming calls arrive at system according to a Poisson process with rate λ. Service times of incoming calls follow the exponential distribution with rate μ₁. If the server is idle upon arrival of an incoming call, it occupies the server. If the server is busy, an incoming call joins the orbit and makes a random delay for exponentially distributed time with parameter σ. From the orbit, an incoming call retries to occupy the server and behaves the same as a primary incoming call. On the other hand, the server makes outgoing calls after some exponentially distributed idle time. We assume that there are several types of outgoing calls whole durations follow distinct exponential distributions with parameter depending on the type of outgoing call. In our system, the server is aim to breakdowns. The duration of periods between breakdowns is exponentially distributed with parameter: γ₀, if the server is idle and γ₁, if the incoming call is in service. The recovery rate is equal to γ₂. We assume that the server cannot be broken while serving outgoing calls due to the fact that in such case the server initiates the service itself. A random process of the number of incoming calls at the system is considered. The aim of the research is to derive stationary probability distribution for this process using limiting distribution of corresponding diffusion process. We derive Kolmogorov system of differential equations and solve it in the limit by σ → 0. The drift coefficient for normalized number of calls in the system (Theorem 1) is obtained. After that, we extend the study to obtain diffusion coefficient and derive the equation for limiting characteristic function of normalized number of calls in the system (Theorem 2). Based on the obtained coefficients, we have built the diffusion approximation of the probability distribution of the number of incoming calls in the system. Numerical example shows high accuracy of the obtained approximation. In this section, we also compare the accuracy of asymptotic-diffusion method and asymptotic analysis method, which gives Gaussian approximation under the same limit condition σ → 0. We show that diffusion approximation have more wide area of applicability.

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